Abstract
In this paper, an asymptotic log-Harnack inequality and some consequent properties are established via the asymptotic coupling method for a class of stochastic 2D hydrodynamical-type systems driven by degenerate noise. The main results are applicable to the stochastic 2D Navier–Stokes equations, stochastic 2D magneto-hydrodynamic equations and stochastic 2D Boussinesq equations, stochastic 2D magnetic Bénard problem, stochastic 3D Leray-\(\alpha \) model and also stochastic shell models of turbulence in the degenerate noise case.
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References
C.T. Anh, N.T. Da, The exponential behaviour and stabilizability of stochastic 2D hydrodynamical type systems, Stochastics, 89 (2017), 593–618.
M. Arnaudon, A. Thalmaier, F.-Y. Wang, Gradient estimates and Harnack inequalities on non-compact Riemannian manifolds, Stochastic Process. Appl. 119 (2009), 3653–3670.
V. Barbu, G. Da Prato, Existence and ergodicity for the two-dimensional stochatsic magneto-hydrodynamics equations, Appl. Math. Optim. 56 (2007), 145–168.
H. Bessaih, B. Ferrario, The regularized 3D Boussinesq equations with fractional Laplacian and no diffusion, J. Differential Equations 262 (2017), 1822–1849.
H. Bessaih, E. Hausenblas, P. A. Razafimandimby, Strong solutions to stochastic hydrodynamical systems with multiplicative noise of jump type, Nonlinear Differ. Equ. Appl. 22 (2015), 1661–1697.
O. Butkovsky, A. Kulik, M. Scheutzow, Generalized couplings and ergodic rates for SPDEs and other Markov models, Ann. Appl. Probab. 30 (2020), 1–39.
J. Bao, F.-Y. Wang, C. Yuan, Asymptotic log-Harnack inequality and applications for Stochastic systems of infinite memory, Stochastic Process. Appl. 129 (2019), 4576–4596.
P. Constantin, N. Glatt-Holtz, V. Vicol, Unique ergodicity for fractionally dissipated stochastically forced 2D Euler equations, Comm. Math. Phys. 330 (2014), 819–857.
I. Chueshov, A. Millet, Stochastic 2D hydrodynamical type systems: well posedness and large deviations, Appl. Math. Optim. 61 (2010), 379–420.
I. Chueshov, A. Millet, Stochastic two-dimensional hydrodynamical systems: Wong-Zakai approximation and support theorem, Stoch. Anal. Appl. 29 (2011), 570–611.
P. Constantin, C. Foias, Navier-Stokes Equations, University of Chicago Press, Chicago, 1988.
G. Da Prato, J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1992.
W. E, J.C. Mattingly, Ergodicity for the Navier-Stokes equation with degenerate random forcing: finite-dimensional approximation, Comm. Pure Appl. Math. 54 (2001), 1386–1402.
W. E, J.C. Mattingly, Y. Sinai, Gibbsian dynamics and ergodicity for the stochastically forced Navier-Stokes equation, Comm. Math. Phys. 224 (2001), 83–106.
J. Földes, S. Friedlander, N. Glatt-Holtz, G. Richards, Asymptotic analysis for randomly forced MHD, SIAM J. Math. Anal. 49 (2017), 4440–4469.
J. Földes, N. Glatt-Holtz, G. Richards, E. Thomann, Ergodic and mixing properties of the Boussinesq equations with a degenerate random forcing, J. Funct. Anal. 269 (2015), 2427–2504.
J. Földes, N. Glatt-Holtz, G. Richards, J.P. Whitehead, Ergodicity in randomly forced Rayleigh-Benard convection, Nonlinearity 29 (2016), 3309–3345.
P.W. Fernando, E. Hausenblas, P.A. Razafimandimby, Irreducibility and exponential mixing of some stochastic hydrodynamical systems driven by pure jump noise, Comm. Math. Phys. 348 (2016), 535–565.
B. Gess, W. Liu, A. Schenke, Random Attractors for Locally Monotone Stochastic Partial Differential Equations, J. Differential Equations 269 (2020), 3414–3455.
N. Glatt-Holtz, J.C. Mattingly, G. Richards, On unique ergodicity in nonlinear stochastic partial differential equations, J. Stat. Phys. 166 (2017), 618–649.
M. Hairer, J.C. Mattingly, Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing, Ann. Math. 164 (2006), 993–1032.
M. Hairer, J.C. Mattingly, A theory of hypoellipticity and unique ergodicity for semilinear stochastic PDEs, Electron. J. Probab. 16 (2011), 658–738.
A. Kulik, M. Scheutzow, Generalized couplings and convergence of transition probabilities, Probab. Theory Relat. Fields 171 (2018), 333–376.
O.A. Ladyzhenskaia, Solution “in the large” of the nonstationary boundary value problem for the Navier-Stokes system with two space variables, Comm. Pure Appl. Math. 12 (1959), 427–433.
S. Li, W. Liu, Y. Xie, Ergodicity of 3D Leray-\(\alpha \)model with fractional dissipation and degenerate stochastic forcing, Infin. Dimens. Anal. Quantum Probab. Relat. Top 22 (2019), 1950002, 20pp.
W. Liu, Harnack inequality and applications for stochastic evolution equations with monotone drifts, J. Evol. Equ. 9 (2009), 747–770.
W. Liu, M. Röckner, SPDE in Hilbert space with locally monotone coefficients, J. Funct. Anal. 259 (2010), 2902–2922.
W. Liu, M. Röckner, Stochastic Partial Differential Equations: An Introduction, Universitext, Springer, 2015.
W. Liu, F.-Y. Wang, Harnack inequality and strong Feller property for stochastic fast-diffusion equations, J. Math. Anal. Appl. 342 (2008), 651-662.
M. Röckner, X. Zhang, Stochastic tamed 3D Navier-Stokes equations: existence, uniqueness and ergodicity, Probab. Theory Related Fields 145 (2009), 211–267.
R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, CBMS-NSF Regional Conference Series in Applied Mathematics, 66 SIAM, Philadelphia, PA, (1995).
F.-Y. Wang, Logarithmic Sobolev inequalities on noncompact Riemannian manifolds, Probab. Theory Related Fields 109 (1997), 417–424.
F.-Y. Wang, Harnack inequality and applications for stochastic generalized porous media equations, Ann. Probab. 35 (2007), 1333–1350.
F.-Y. Wang, Harnack inequalities on manifolds with boundary and applications, J. Math. Pures Appl. 94 (2010), 304-321.
F.-Y. Wang, Harnack inequalities and Applications for Stochastic Partial Differential Equations, Springer, Berlin, 2013.
F.-Y. Wang, T.S. Zhang, Log-Harnack inequality for mild solutions of SPDEs with multiplicative noise, Stochastic Process. Appl. 124 (2014), 1261–1274.
L. Xu, A modified log-Harnack inequality and asymptotically strong Feller property, J. Evol. Equ. 11 (2011), 925–942.
J. Yang, J. Zhai, Asymptotics of stochastic 2D hydrodynamical type systems in unbounded domains, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 20 (2017), 1750017, 25pp, 25pp.
Acknowledgements
The authors would like to thank the referees for their very valuable suggestions and constructive comments. This work is supported by NSFC (Nos. 11822106, 11831014, 11571147) and the PAPD of Jiangsu Higher Education Institutions.
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Hong, W., Li, S. & Liu, W. Asymptotic log-Harnack inequality and applications for stochastic 2D hydrodynamical-type systems with degenerate noise. J. Evol. Equ. 21, 419–440 (2021). https://doi.org/10.1007/s00028-020-00587-w
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DOI: https://doi.org/10.1007/s00028-020-00587-w
Keywords
- Hydrodynamical systems
- Asymptotic log-Harnack inequality
- Asymptotically strong Feller
- Degenerate noise
- Ergodicity